Wal-Mart, a discount store chain, is planning to build a new store in
Rock Springs, Maryland. The parcel of land the company owns is large
enough to accommodate a store with 140,000 square feet of floor space.
Based on marketing and demographic surveys of the area and historical
data from its other stores, Wal-Mart estimates its annual profit
contribution per square foot for each of the store's departments to be
as shown in the following table.
Department Profit contribution per ft2
Men's clothing $4.25
Women's clothing $5.10
Children's clothing $4.50
Toys $5.20
Housewares $4.10
Electronics $4.90
Auto supplies $3.80
Each department must have at least 15,000 ft2 of floor space and no
department can have more than 20% of the total retail floor space. Men's
women's and children's clothing plus housewares keep all their stock on
the retail floor; however, toys, electronics, and auto supplies keep
some items (bicycles, televisions, tires, etc.) in inventory. Thus, 10%
of the total retail floor space devoted to these three departments must
be set aside outside the retail area for stocking inventory.
Formulate a linear programming model that can be used to determine
the floor space that should be devoted to each department in order to
maximize profit contribution.
To formulate a linear programming model for this problem, we need to define the decision variables, objective function, and constraints.
1. Decision Variables:
Let's denote the floor space allocated to each department as follows:
- x1: Men's clothing floor space (in ft2)
- x2: Women's clothing floor space (in ft2)
- x3: Children's clothing floor space (in ft2)
- x4: Toys floor space (in ft2)
- x5: Housewares floor space (in ft2)
- x6: Electronics floor space (in ft2)
- x7: Auto supplies floor space (in ft2)
2. Objective Function:
The objective is to maximize the total profit contribution. The profit contribution from each department is given per square foot. So, the objective function can be written as:
Maximize Z = 4.25x1 + 5.10x2 + 4.50x3 + 5.20x4 + 4.10x5 + 4.90x6 + 3.80x7
3. Constraints:
We need to consider the following constraints:
- Each department must have at least 15,000 ft2 of floor space:
x1 ≥ 15,000
x2 ≥ 15,000
x3 ≥ 15,000
x4 ≥ 15,000
x5 ≥ 15,000
x6 ≥ 15,000
x7 ≥ 15,000
- No department can have more than 20% of the total retail floor space:
x1 + x2 + x3 + x4 + x5 + x6 + x7 ≤ 0.2 * (x1 + x2 + x3 + x4 + x5 + x6 + x7)
- Inventory space constraint (10% of total retail floor space for toys, electronics, and auto supplies):
0.1 * (x4 + x6 + x7) = x4 + x6 + x7 - (x4 + x6 + x7)
- Total floor space should not exceed 140,000 ft2:
x1 + x2 + x3 + x4 + x5 + x6 + x7 ≤ 140,000
All the variables are non-negative.
This formulation represents a linear programming model that can be used to determine the optimal floor space allocation in order to maximize profit contribution for Wal-Mart's new store in Rock Springs, Maryland.